Percentiles and Quartiles

The mean and median tell you about the center of a dataset, but they don’t tell you where a specific value stands relative to all the others. Percentiles and quartiles answer a different question: What percentage of values fall below this point?

What Is a Percentile?

A percentile indicates the percentage of values in a dataset that fall at or below a given value.

• If you score in the 90th percentile on a test, 90% of test-takers scored at or below your score.
• The 50th percentile is the median — half the values are below it, half above.
• A higher percentile means a higher relative position in the dataset.

Formula to find the percentile rank of a value:

$\text{Percentile Rank} = \frac{\text{Number of values below } x}{\text{Total number of values}} \times 100$

Example 1: Test Score Percentile

25 students took a test. Their scores, sorted in order:

$52, 58, 61, 63, 65, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 84, 85, 87, 89, 91, 93, 95, 98$

A student scored 84. What percentile is that?

Step 1: Count how many values are below 84.

Looking at the sorted list, the values below 84 are: 52, 58, 61, 63, 65, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82. That’s 17 values.

Step 2: Apply the formula.

$\text{Percentile Rank} = \frac{17}{25} \times 100 = 68\text{th percentile}$

Answer: A score of 84 is at the 68th percentile — this student scored higher than 68% of the class.

Quartiles

Quartiles divide a sorted dataset into four equal parts. They are special percentiles:

How to Find Quartiles

Step 1: Sort the data from smallest to largest.

Step 2: Find the median ($Q_2$) — this splits the data into a lower half and an upper half.

Step 3: Find $Q_1$ — the median of the lower half (values below $Q_2$).

Step 4: Find $Q_3$ — the median of the upper half (values above $Q_2$).

Example 2: Finding Quartiles

Dataset (already sorted): $12, 15, 18, 22, 25, 28, 30, 35, 40, 42, 48$

There are 11 values.

$Q_2$ (Median): The middle value is the 6th value.

$Q_2 = 28$

Lower half (values below 28): $12, 15, 18, 22, 25$ (5 values)

$Q_1$: The median of the lower half is the 3rd value.

$Q_1 = 18$

Upper half (values above 28): $30, 35, 40, 42, 48$ (5 values)

$Q_3$: The median of the upper half is the 3rd value.

$Q_3 = 40$

The Five-Number Summary

The five-number summary provides a complete picture of how data is distributed:

These five numbers are the foundation of a box plot (also called a box-and-whisker plot). The box spans from $Q_1$ to $Q_3$, a line inside the box marks the median, and whiskers extend to the minimum and maximum (or to the fences, if there are outliers).

Interquartile Range (IQR)

The interquartile range measures the spread of the middle 50% of the data:

$\text{IQR} = Q_3 - Q_1$

From Example 2:

$\text{IQR} = 40 - 18 = 22$

The IQR is more resistant to outliers than the range (max minus min) because it ignores the extreme values at both ends.

Identifying Outliers with the IQR

A common rule for identifying outliers uses the IQR:

$\text{Lower fence} = Q_1 - 1.5 \times \text{IQR}$

$\text{Upper fence} = Q_3 + 1.5 \times \text{IQR}$

Any value below the lower fence or above the upper fence is considered an outlier.

Example 3: Detecting Outliers

Dataset (sorted): $3, 15, 18, 22, 25, 28, 30, 35, 40, 42, 85$

Step 1: Find the quartiles. There are 11 values.

$Q_2 = 28 \quad \text{(6th value)}$

Lower half: $3, 15, 18, 22, 25$ $\Rightarrow$ $Q_1 = 18$

Upper half: $30, 35, 40, 42, 85$ $\Rightarrow$ $Q_3 = 40$

Step 2: Calculate the IQR.

$\text{IQR} = 40 - 18 = 22$

Step 3: Calculate the fences.

$\text{Lower fence} = 18 - 1.5 \times 22 = 18 - 33 = -15$

$\text{Upper fence} = 40 + 1.5 \times 22 = 40 + 33 = 73$

Step 4: Check for values outside the fences.

• $3$ is above $-15$ — not an outlier.
• $85$ is above $73$ — outlier.

Answer: The value 85 is an outlier. The value 3, while the lowest, is within the fences and is not an outlier by this rule.

Real-World Application: Nursing — Growth Chart Percentiles

Pediatric nurses use percentile charts to track children’s growth. When a parent hears “your baby is in the 75th percentile for weight,” here is what that means:

A 6-month-old boy weighs 18.5 lbs. The growth chart data for 6-month-old boys shows:

Interpretation: At 18.5 lbs, this baby falls just below the 75th percentile — approximately 75% of 6-month-old boys weigh less. This is a healthy weight.

Why percentiles matter more than raw values:

• A baby consistently at the 75th percentile across checkups is growing normally.
• A baby who drops from the 75th percentile to the 25th percentile over several visits is a concern — not because the 25th percentile is “bad,” but because a large shift in percentile indicates a change in growth pattern.
• Percentiles are always relative to the reference population. The 50th percentile is not the “target” — consistent tracking along any percentile line indicates healthy growth.

Calculating the IQR for this data:

$\text{IQR} = Q_3 - Q_1 = 18.8 - 15.9 = 2.9 \text{ lbs}$

This tells us the middle 50% of 6-month-old boys weigh within a 2.9 lb range. A weight outside $Q_1 - 1.5 \times 2.9 = 11.55$ lbs or $Q_3 + 1.5 \times 2.9 = 23.15$ lbs would be flagged for further evaluation.

Percentile and Quartile Reference

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Percentiles tell you what percentage of values fall below a given point — the 90th percentile means 90% scored lower
• Quartiles divide data into four equal parts: $Q_1$ (25th), $Q_2$ (median, 50th), and $Q_3$ (75th)
• The five-number summary (min, $Q_1$, median, $Q_3$, max) gives a complete snapshot of a distribution
• IQR ($Q_3 - Q_1$) measures the spread of the middle 50% and is resistant to outliers
• Use the 1.5 $\times$ IQR rule to identify outliers: values below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$
• In real-world applications like growth charts, consistency of percentile matters more than the percentile number itself

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